Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Here we will discuss about one of the important geometrical property of parallelogram.

A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel

Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR.

To prove: PQRS is a parallelogram.

Construction: Join PR and QS such that they intersect at O.

Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Proof:

          Statement

          Reason

In ∆OPQ and ∆ORS,

1. ∠OPQ = ∠ORS


1. PQ ∥ SR and PR is a transversal.

2. ∠POQ = ∠ROS

2. Opposite angles are equal.

3. PQ = RS

3. Given.

4. ∆OPQ ≅ ∆ORS

Therefore, OP = OR, OQ = OS.

In ∆OPS and ∆OQR,

4. By AAS criterion of congruency.

CPCTC


5. OP = OC, OQ = OS, ∠POS = ∠QOR

5. By statement 4 and reason 2.

6. ∆OPS ≅ ∆OQR

Therefore, PS = QR, ∠OPS= ∠ORQ

6. By SAS criterion of congruency.

CPCTC

7. PS ∥QR.

7. Alternate angles are equal.

8. PQRS is a parallelogram (Proved).

8. PQ ∥ SR and statement 7.

Corollary: In a parallelogram, each pair of opposite sides are parallel as well as equal.





9th Grade Math

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